A210565 Triangle of coefficients of polynomials u(n,x) jointly generated with A210595; see the Formula section.

1, 2, 2, 3, 5, 3, 4, 9, 10, 5, 5, 14, 22, 20, 8, 6, 20, 40, 51, 38, 13, 7, 27, 65, 105, 111, 71, 21, 8, 35, 98, 190, 256, 233, 130, 34, 9, 44, 140, 315, 511, 594, 474, 235, 55, 10, 54, 192, 490, 924, 1295, 1324, 942, 420, 89, 11, 65, 255, 726, 1554, 2534, 3130, 2860, 1836, 744, 144

Offset

1,2

Comments

Row n starts with n and ends with F(n+1), where F=A000045 (Fibonacci numbers).

Row sums: A005409.

Alternating row sums: 1,0,1,0,1,0,1,0,1,0,1,0, ...

For a discussion and guide to related arrays, see A208510.

Links

G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Formula

u(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x) + 1,

v(n,x) = x*u(n-1,x) + v(n-1,x) + 1,

where u(1,x) = 1, v(1,x) = 1.

T(n, k) = [x^k]( u(n, x) ), where u(n, x) = (1+x)*u(n-1,x) + x^2*u(n-2,x) + 1 + x, u(1, x) = 1, and u(2, x) = 2 + 2*x. - G. C. Greubel, May 24 2021

Example

First five rows:
  1;
  2,  2;
  3,  5,  3;
  4,  9, 10,  5;
  5, 14, 22, 20, 8;
First three polynomials u(n,x):
u(1, x) = 1;
u(2, x) = 2 + 2*x;
u(3, x) = 3 + 5*x + 3*x^2.

Mathematica

(* First program *)
u[1, x_]:= 1; v[1, x_]:= 1; z = 16;
u[n_, x_]:= x*u[n-1, x] + (x+1)*v[n-1, x] + 1;
v[n_, x_]:= x*u[n-1, x] + v[n-1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210565 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210595 *)
(* Second program *)
u[n_, x_]:= u[n, x]= If[n<2, (n+1)*(1+x)^n, (1+x)*u[n-1, x] +x^2*u[n-2, x] +1+x];
T[n_]:= CoefficientList[Series[u[n, x], {x, 0, n}], x];
Table[T[n-1], {n, 12}] (* G. C. Greubel, May 23 2021 *)

Prog

(Sage)
@CachedFunction
def u(n, x): return (n+1)*(1+x)^n if (n<2) else (1+x)*u(n-1, x) + x^2*u(n-2, x) +1+x
def T(n): return taylor( u(n, x) , x, 0, n).coefficients(x, sparse=False)
flatten([T(n-1) for n in (1..12)]) # G. C. Greubel, May 23 2021

Crossrefs

Cf. A208510, A210595.

Sequence in context: A295120 A196957 A124727 * A125101 A208519 A336725

Adjacent sequences: A210562 A210563 A210564 * A210566 A210567 A210568

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 23 2012

Status

approved